1. **Problem Statement:** We will learn about Measures of Central Tendency and Dispersion for grouped and ungrouped data, including how to calculate mean, identify modal and median intervals, understand range and percentiles, and use five-number summaries and graphs. 2. **Mean Calculation:** - For ungrouped data, mean is the sum of all values divided by the number of values: $$\text{Mean} = \frac{\sum x_i}{n}$$ where $x_i$ are data points and $n$ is the count. - For grouped data, use midpoints ($m_i$) of class intervals and frequencies ($f_i$): $$\text{Mean} = \frac{\sum f_i m_i}{\sum f_i}$$ 3. **Modal Interval and Median Interval:** - The modal interval is the class interval with the highest frequency. - The median interval is the class interval where the cumulative frequency reaches or exceeds half the total frequency. 4. **Range and Measures of Dispersion:** - Range = Maximum value - Minimum value. - Percentiles divide data into 100 equal parts; quartiles divide data into 4 equal parts. - Inter-quartile range (IQR) = $Q_3 - Q_1$ measures middle 50% spread. - Semi-inter-quartile range = $\frac{IQR}{2}$. 5. **Five Number Summary:** - Consists of Minimum, $Q_1$, Median ($Q_2$), $Q_3$, and Maximum. - Used to create box and whisker diagrams showing data spread and outliers. 6. **Using Statistical Summaries and Graphs:** - Analyze data by comparing means, medians, modes, and dispersion. - Use bar graphs for categorical data, line graphs for trends over time, and histograms for frequency distribution. 7. **Summary:** - These tools help understand data distribution, central values, and variability. - Graphs visually represent data for easier interpretation and decision making.